“It is He Who made the sun to be a shining glory, and the moon to be a light of beauty, and measured out stages for it, that you might know the number of years and the count of time. Allah did not create this except in truth and righteousness. And He explains His signs in detail, for those who understand” (Quran 10:5)
It is Sunnah to fast whatever one can of the month of Muharram, because of the hadeeth: “The best of fasting after Ramadaan is the month of Allaah Muharram, and the best of prayer after the obligatory prayers is prayer at night (qiyaam al-layl).” (Muslim, 1163).
It is highly recommended to fast on the day of ‘Aashoora’ which is the tenth of Muharram.
It was narrated that Ibn ‘Abbaas (may Allaah be pleased with him) was asked about fasting on the day of ‘Aashoora’. He said, “I do not know of any day on the Messenger of Allaah (peace and blessings of Allaah be upon him) fasted that was better than this day and any month that was better than this month, meaning Ramadaan.” (al-Bukhaari, 2006; Muslim, 1132).
It is Sunnah to fast the day before or the day after along with ‘Aashoora’, in order to be different from the Jews.
May Allah give us success.